# Waiting time in M/M/n queue

I was able to numerically "show" that an M/M/n queue is concave increasing in the number for $$n$$ servers, which makes intuitive sense. However is there a formal analytical proof for it? Where can I find it?

Dyer and Proll (1977)1 showed that for an M/M/c queue, the mean waiting time is a strictly decreasing and convex function of c.

Reference

[1] Dyer, M. E., Proll, L. G. (1977). On the Validity of Marginal Analysis for Allocating Servers in M/M/c Queues. Management Science. 23(9):1019-1022.

• For a broader overview see Yu et al. (2006). May 17 '20 at 19:54
• Thank you so much.
– Paul
May 18 '20 at 9:52

You can derive them from the balance equations. If you check Taha's or Lieberman's Introduction to OR books, you can find the proofs.